The tensor-product conjecture for torsion subalgebras of integral-dynamics Kirchberg algebras

Let $S\subset\mathbb{N}^{\times}\setminus\{1\}$ be a family of relatively prime numbers with $\lvert S\rvert\geq 2$. Let $\mathcal{A}_S$ be the torsion subalgebra associated to $S$, let $\mathcal{Q}_S$ be the corresponding unital UCT Kirchberg algebra, and let $g_S$ denote the greatest common divisor of $\{p-1:p\in S\}$. For an index set of the appropriate cardinality, write $e_1=(\delta_{1,j})_j\in(\mathbb{Z}/g_S\mathbb{Z})^{2^{\lvert S\rvert-2}}$. Tensor-product conjecture. The algebra $\mathcal{A}_S$ is isomorphic to

$$\bigotimes_{p\in S}\mathcal{O}_p.$$

Equivalently, $\mathcal{Q}_S$ is the unital UCT Kirchberg algebra with

$$(K_0(\mathcal{Q}_S),[1],K_1(\mathcal{Q}_S))=\left(\mathbb{Z}^{2^{\lvert S\rvert-1}}\oplus(\mathbb{Z}/g_S\mathbb{Z})^{2^{\lvert S\rvert-2}},(0,e_1),\mathbb{Z}^{2^{\lvert S\rvert-1}}\oplus(\mathbb{Z}/g_S\mathbb{Z})^{2^{\lvert S\rvert-2}}\right).$$

In particular, for non-empty sets $S,T\subset\mathbb{N}^{\times}\setminus\{1\}$ of relatively prime numbers, $\mathcal{Q}_S$ is isomorphic to $\mathcal{Q}_T$ if and only if $\lvert S\rvert=\lvert T\rvert$ and $g_S=g_T$. The conjecture extends the cases already established for $\lvert S\rvert\leq 2$ and $g_S=1$; the general case with $\lvert S\rvert\geq 3$ and $g_S>1$ is left open.